6th St.Petersburg Workshop on Simulation (2009) 419-423
On a Price-Liquidity Threshold Regime Swicthing
Model1
Nelson S. Rianço2 , Manuel L. Esquı́vel3 , Pedro P. Mota4 ,
Carlos A. Veiga5
Abstract
We propose a model for the price evolution of stock exchange assets that
incorporates the information contained in liquidity values as expressed in
local currency. The model is given by a system of stochastic differential
equations, one for price and another for liquidity, having regime switching
parameters that change according to the crossings of thresholds by the trajectories of the processes. By means of a simple simulation study we present
some of the properties of the model and show that it allows to recover some
of the evolution features of a typical stock of the equity Portuguese market.
1. Introduction
Liquidity in equity markets may be defined by several concurrent quantities directly observed. The most discussed in the literature are the bid-ask spread (see
[3] and [7]) and the volume of transactions ([4], [2], [5] and references therein).
A number of empirical studies, referenced in the studies quoted, show the reciprocal influence of liquidity on price levels and of price levels on diverse liquidity
measures.
We will report elsewhere a statistical analysis of both the bid-ask spread and
the volume of transactions, for a large set of securities beside the one studied in this
paper, showing that the quantity having the best statistical properties allowing a
log-normal model is the volume of transaction as expressed in local currency.
1
This work was partially supported by Financiamento Base 2008 ISFL-1-297 from
FCT/MCTES/PT.
2
Santander Totta, Portugal, E-mail: [email protected]
3
Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa (FCT/UNL), Portugal, E-mail: [email protected]
4
FCT/UNL, Portugal, E-mail: [email protected]
5
Millenniumbcp, Portugal, E-mail: [email protected]
2. The data set
The data represented in figure 1 exhibits remarkable features. In the price evolution several drift regimes are readily noticed. In the liquidity evolution there is
very strong volatility as well as a small number of outliers.
Figure 1: The price and liquidity data for Portugal-Telecom (PT).
Supposing both price and volume to be log-normal we have estimated the
parameters in six different periods corresponding to dates from 1 to 410, 411 to
458,459 to 730, 731 to 828, 829 to 1226 and 1227 to 1453. The results are presented
in the next table.
Estimated parameters in lognormal price and liquidity
Pri.
Liq.
µt
σt
νt
ρt
1–410
411–458
459–730
731–828
829–1226
1227-1453
0.002417
0.015352
−0.013253
0.037116
0.001351
0.02083
0.008005
0.027699
−0.001941
0.026868
0.00006
0.020535
1–410
411–458
459–730
731–828
829–1226
1227-1453
0.200038
0.629027
0.224623
0.660619
0.252027
0.716109
0.130697
0.50752
0.149504
0.541546
0.144188
0.542163
Our preliminary observation of the price evolution is confirmed by the different
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values obtained in the estimated µt . The estimated parameters for the liquidity
clearly show two different regimes affecting both νt and ρt ; the first one, from date
1 to date 730, has clearly more intense volatility and drift.
3. The Model
We propose a price-volume coupled model with regimes and thresholds given by

dPt


= µt dt + σt dBt1 , P0 ∈ R+

Pt
(1)

 dLt = κ (θ − log(L ))dt + ρ dB 2 , L ∈ R+

t t
t
t
0
t
Lt
where (Bt1 )t≥0 and (Bt2 )t≥0 are Brownian processes with variance-covariance matrix
·
¸
1 ρ
Σ
\=
ρ 1
We remark that in between the random times of threshold crossing both the price
and the liquidity are log-normal. The interplay of the regimes and the threshold is
given by the following. The thresholds for the price and liquidity being denoted,
respectively, by Pm , PM , Lm , LM , the coefficients σt , ρt , κt and θt satisfy relations
similar to this one satisfied by µt :

h

if Lt > PM
µ
µt = µ
if Pm ≤ Lt ≤ PM

 d
µ
if Lt < Pm
where µh , µ and µd are real numbers (we may have µh > 0 and µd = −µh or
another sign combination). In fact, combining different signs for the drift parameters, this model allows for, at least, 16 scenarios that we detail in the following
tables.
Scenarios
Liquidity on highest price threshold
Liquidity on lowest price threshold
Price on highest liquidity threshold
Price on lowest liquidity threshold
Scenarios
Liq. on highest price threshold
Liq. on lowest price threshold
Pri. on highest liquidity threshold
Pri. on lowest liquidity threshold
IX
↓
↑
↑
↑
I
↑
↓
↓
↑
X
↑
↓
↓
↓
II
↓
↓
↓
↓
XI
↑
↓
↑
↓
III
↓
↓
↓
↑
XII
↑
↓
↑
↑
IV
↓
↓
↑
↓
V
↓
↓
↑
↑
XIII
↑
↑
↓
↓
VI
↓
↑
↓
↓
XIV
↑
↑
↓
↑
VII
↓
↑
↓
↑
XV
↑
↑
↑
↓
VIII
↓
↑
↑
↓
XVI
↑
↑
↑
↑
Of course, some of these scenarios may lack economic sense. The first scenario
which we consider to be the most plausible, may be describe the first scenario in
the following way.
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1. If the price gets bigger than the highest threshold then liquidity has a tendency to go up.
2. If the price gets smaller than the lowest threshold then liquidity has a tendency to go down.
3. If liquidity gets bigger than the highest threshold then the price has a tendency to go down.
4. If the liquidity gets smaller than the lowest threshold then the price has a
tendency to go up.
The numerical integration of the model without noise and tight thresholds exhibits an almost periodic solution behavior. This feature clearly differentiates the
threshold regime switching model here proposed from an alternative of a predatorprey model with additive noise and correspondent non random volatility. Such a
model has always a periodic solution if the noise is set to be null (see [1]).
Price
4.326
Liquidity
6
2.1822·10
Price & Liquidity Phase Plane
6
2.1822·10
4.325
4.3214.3224.3234.3244.3254.326
500 100015002000250030003500
4.324
6
6
2.1818·10
2.1818·10
4.323
4.322
6
6
2.1816·10
4.321
2.1816·10
6
2.1814·10
6
2.1814·10
500 1000 1500 2000 2500 3000 3500
Figure 2: Price, liquidity and phase plane simulation outputs for model scenario I
with null noise built with PT data estimated parameters.
A simulation of the first scenario mentioned above was implemented using the
Euler scheme. A sample of the outputs are presented in figure 3.
The simulations show that the model allows to recover the general form for the
price evolution as given by figure 1. As for the volume evolution we notice that
we can’t reproduce the small set of dates with very high values.
4. Estimation procedure
For each vector of plausible thresholds (Pm , PM , Lm , LM ) ∈ R4 we split the price
and the liquidity in sets corresponding to different regimes. Looking at the price
data and for the fixed thresholds, we know that when the price is below the lower
threshold (Pm ) we should consider the corresponding liquidity in the first regime,
when the price is betwwen the lower and the upper threshold (PM ) we consider
the liquidity data in the second regime and when the price is above the upper
threshold we consider the data in the third regime. Loooking at the liquidity we
can do the same in order to split the price data in three regimes. Next, for the
price we can estimate the drift and the volatility parameters in each of the regimes
422
Price
Liquidity
4.36
6
2.3·10
4.35
4.34
6
2.2·10
4.33
6
4.32
2.1·10
4.31
200 400 600 800 1000 1200 1400
200 400 600 800 1000 1200 1400
Price
Liquidity
4.34
6
2.3·10
4.33
6
2.25·10
4.32
6
2.2·10
6
4.31
2.15·10
200 400 600 800 1000 1200 1400
200 400 600 800 1000 1200 1400
6
2.05·10
4.29
Price
Liquidity
4.33
6
2.3·10
4.32
6
2.2·10
4.31
6
2.1·10
200 400 600 800 1000 1200 1400
200 400 600 800 1000 1200 1400
4.29
Figure 3: Three price liquidity simulation outputs for model scenario I with PT
data estimated parameters.
and then we compute the sum of the square of the residuals
LSPn =
3 n−1
X
X³
Si+1 − E[Si+1 |Si ]1Si ∈Regime(j)
´2
.
(2)
j=1 i=0
Finally, we choose for the liquidity threshold estimates the values that minimizes
the previous function. Repeating the same computations for the liquidity we
estimate the thresholds for the prices.
5. Conclusions and further work
The threshold regime switching model here investigated recovers some of the general traits of the joint price-liquidity evolution for a typical stock market security
of the Portuguese market. Existence, stability, and asymptotic properties of the
model will be presented in a longer version of this work. The consistency of the
model threshold and diffusion parameters estimators proposed above remain to be
proved but some work for a more simple regime switching threshold model (see [6])
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indicates that this may be a difficult problem. To determine the practical relevance
of incorporating the liquidity information on the price evolution the benchmarking
of this model against classical log-normal models for the price will be performed.
References
[1] Abundo M. (1991) A stochastic model for predator-prey systems: basic properties, stability and computer simulation. Journal of Mathematical Biology 9,
6, 495–511.
[2] Johnson T. C. (2007) Volume, liquidity, and liquidity risk., Journal of Financial Economics. 87 , 2, 388–418.
[3] Glosten, L. & Milgrom, P. (1985) Bid, Ask and Transaction Prices in a Specialist Market with heterogeneously informed traders. Journal of Financial
Economics. 14 71–100.
[4] Lee, J. & Kim, S. (2008) Numerical Solutions Of Option Pricing Model With
Liquidity Risk.,Communications of the Korean Mathematical Society. 23 , 1,
141–151.
[5] Malinova, K. & Park, A. (2008) Trading Volume in Dealer Markets, preprint
University of Toronto.
[6] Mota, P. P. (2008) Brownian Motion with Drift Threshold Model. PhD dissertation FCT/UNL.
[7] Yakov, A. & Mendelson, H. & Pedersen, L. H. (2006) Liquidity and asset prices.
Foundations and Trends in Finance. 1, 4, 269–364.
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